A box contains 4 red cards, 6 blue cards, and 10 green cards. What is the smallest number of cards we must remove from the box to make sure that we have 2 cards of the same color?

To determine the smallest number of cards we must remove to ensure that we have 2 cards of the same color, we can use the pigeonhole principle.

The pigeonhole principle states that if we have more pigeons than pigeonholes, there must be at least one pigeonhole with more than one pigeon.

In this context, the pigeons are the cards, and the pigeonholes are the colors.

Since we have 4 red cards, 6 blue cards, and 10 green cards, we have a total of 20 cards. However, we are trying to find the smallest number of cards we must remove to guarantee having 2 cards of the same color.

To minimize the number of cards we remove, we can assume that we remove all the red and blue cards, leaving only the green cards in the box.

Thus, we would need to remove all 4 red cards and all 6 blue cards.

Therefore, the smallest number of cards we must remove from the box to make sure we have 2 cards of the same color is 10 (4 red cards + 6 blue cards).

To find the smallest number of cards we must remove to make sure we have 2 cards of the same color, we can use the Pigeonhole Principle.

The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.

In this scenario, the cards are the pigeons and the colors (red, blue, and green) are the pigeonholes. We want to find the smallest number of cards we need to remove to ensure that we have 2 cards of the same color.

The worst-case scenario is that we remove all cards of two different colors, leaving us with only one color. Let's consider the case where we remove all the red cards and all the blue cards. In this case, we will be left with only the green cards. However, we still don't have two cards of the same color.

Therefore, we need to remove at least one more card to ensure that we have two cards of the same color. We can remove any card, and regardless of its color, we will have at least two cards of the same color.

So, the smallest number of cards we must remove from the box to make sure that we have 2 cards of the same color is 2.

There are only 3 colors.

Any group of more than 3 cards will have to have a duplicate color.